I use mathematics as a tool to explain how things work. My research focuses on the use of concepts of geometry and analysis to understand physical problems. The analysis of this hidden "geometrical world" gives new insight into concrete problems -- an insight that is not accessible by direct experimental or numerical inspection. In particular, I use mathematical methods to predict and explain the complex motion of various dynamical systems, such as satellites, asteroids, electric circuits, and fluids. Many mechanical problems are geometrical in nature, although in a higher-dimensional space -- the so-called phase space -- than the usual three-dimensional physical space. Some fascinating phenomena, such as the stable levitation of charged particles in an oscillating electric field, are much better explained when translated into an equivalent geometrical setting. This combined approach has allowed to predict a new phenomenon in the motion of a chain of interconnected pendula -- a system that imitates the current in superconducting crystals. This new observation was later rediscovered by experimentalists.